| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Session | Specimen |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of binomial distributions |
| Type | One-tailed test critical region |
| Difficulty | Challenging +1.2 This is a Further Statistics 1 question covering hypothesis testing with binomial and geometric distributions, including critical regions, significance levels, and power functions. While it requires understanding of multiple concepts (binomial/geometric distributions, one-tailed tests, power functions), the calculations are mostly standard applications of formulas with some algebraic manipulation. The power function derivation in part (e)(i) is shown rather than derived from scratch, and part (f) requires interpretation rather than complex reasoning. This is moderately above average difficulty due to the breadth of concepts and the power function component, but remains a structured multi-part question with clear pathways. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02f Geometric distribution: conditions5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(X \sim B(20, 0.2)\) and seek \(c\) such that \(P(X \leqslant c) < 0.10\) | M1 | 3.3 - Realising the need to use \(B(20,0.2)\) with method for finding CR or implied by a correct CR |
| \([P(X \leqslant 1) = 0.0692]\) CR is \(X \leqslant 1\) | A1 | 1.1b - \(X \leqslant 1\) or \(X < 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Size \(= \mathbf{0.0692}\) | B1ft | 1.2 - awrt 0.0692 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(Y =\) no. of spins until red obtained so \(Y \sim \text{Geo}(0.2)\) | M1 | 3.3 - Realising that the model \(\text{Geo}(0.2)\) is needed |
| \(\mu = \frac{1}{p}\) so if \(p < 0.2\) then mean is larger so seek \(d\) so that \(P(Y \geqslant d) < 0.10\) | M1 | 2.4 - Realising the key step that they need to find \(P(Y \geqslant d) < 0.10\) |
| \(P(Y \geqslant d) = (0.8)^{d-1}\) | M1 | 3.4 - Using the model \((0.8)^{d-1}\) |
| \((0.8)^{d-1} < 0.10 \Rightarrow d-1 > \frac{\log(0.1)}{\log(0.8)}\) | M1 | 1.1b - Using \((0.8)^{d-1}<0.10\) and finding a method to solve |
| \(d > 11.3\ldots\) | A1 | 1.1b - For \(d > 11.3\) |
| CR is \(Y \geqslant 12\) | A1 | 2.2b - For \(Y \geqslant 12\) or \(Y > 11\) (a correct inference) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Size \(= [0.8^{11} = 0.085899\ldots] = \mathbf{0.0859}\) | B1 | 1.1b - awrt 0.0692; ft their answer to part (c) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Power \(= P(\text{reject } H_0 \text{ when false}) = P(X \leqslant 1 \mid X \sim B(20,p))\) | M1 | 2.1 - Using \(B(20,p)\) and realising they need to find \(P(X \leqslant 1)\) |
| \(= (1-p)^{20} + 20(1-p)^{19}p\) | M1 | 1.1b - Using \(P(X=0)+P(X=1)\) |
| \(= (1-p)^{19}(1+19p)\) | A1*cso | 1.1b - Fully correct proof (no errors) cso |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Power \(= (1-p)^{11}\) | B1 | 1.1b - For \((1-p)^{11}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Sam's test has smaller \(P(\text{Type I error})\) (or size) so is better | B1 | 2.2a - Making a deduction about the tests using answers to parts (b) and (d) |
| Power of Sam's test \(= 0.1755\ldots\) | B1 | 1.1b - awrt 0.0176 |
| Power of Tessa's test \(= 0.85^{11} = 0.1673\ldots\) | B1 | 1.1b - awrt 0.167 |
| So for \(p = 0.15\) Sam's test is recommended | B1 | 2.2b - A correct inference about which test is recommended |
# Question 7:
## Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $X \sim B(20, 0.2)$ and seek $c$ such that $P(X \leqslant c) < 0.10$ | M1 | 3.3 - Realising the need to use $B(20,0.2)$ with method for finding CR or implied by a correct CR |
| $[P(X \leqslant 1) = 0.0692]$ CR is $X \leqslant 1$ | A1 | 1.1b - $X \leqslant 1$ or $X < 2$ |
## Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Size $= \mathbf{0.0692}$ | B1ft | 1.2 - awrt 0.0692 |
## Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $Y =$ no. of spins until red obtained so $Y \sim \text{Geo}(0.2)$ | M1 | 3.3 - Realising that the model $\text{Geo}(0.2)$ is needed |
| $\mu = \frac{1}{p}$ so if $p < 0.2$ then mean is larger so seek $d$ so that $P(Y \geqslant d) < 0.10$ | M1 | 2.4 - Realising the key step that they need to find $P(Y \geqslant d) < 0.10$ |
| $P(Y \geqslant d) = (0.8)^{d-1}$ | M1 | 3.4 - Using the model $(0.8)^{d-1}$ |
| $(0.8)^{d-1} < 0.10 \Rightarrow d-1 > \frac{\log(0.1)}{\log(0.8)}$ | M1 | 1.1b - Using $(0.8)^{d-1}<0.10$ and finding a method to solve |
| $d > 11.3\ldots$ | A1 | 1.1b - For $d > 11.3$ |
| CR is $Y \geqslant 12$ | A1 | 2.2b - For $Y \geqslant 12$ or $Y > 11$ (a correct inference) |
## Part (d):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Size $= [0.8^{11} = 0.085899\ldots] = \mathbf{0.0859}$ | B1 | 1.1b - awrt 0.0692; ft their answer to part (c) |
## Part (e)(i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Power $= P(\text{reject } H_0 \text{ when false}) = P(X \leqslant 1 \mid X \sim B(20,p))$ | M1 | 2.1 - Using $B(20,p)$ and realising they need to find $P(X \leqslant 1)$ |
| $= (1-p)^{20} + 20(1-p)^{19}p$ | M1 | 1.1b - Using $P(X=0)+P(X=1)$ |
| $= (1-p)^{19}(1+19p)$ | A1*cso | 1.1b - Fully correct proof (no errors) cso |
## Part (e)(ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Power $= (1-p)^{11}$ | B1 | 1.1b - For $(1-p)^{11}$ |
## Part (f):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Sam's test has smaller $P(\text{Type I error})$ (or size) so is better | B1 | 2.2a - Making a deduction about the tests using answers to parts (b) and (d) |
| Power of Sam's test $= 0.1755\ldots$ | B1 | 1.1b - awrt 0.0176 |
| Power of Tessa's test $= 0.85^{11} = 0.1673\ldots$ | B1 | 1.1b - awrt 0.167 |
| So for $p = 0.15$ **Sam's test** is recommended | B1 | 2.2b - A correct inference about which test is recommended |\begin{enumerate}
\item Sam and Tessa are testing a spinner to see if the probability, p , of it landing on red is less than $\frac { 1 } { 5 }$. They both use a $10 \%$ significance level.
\end{enumerate}
Sam decides to spin the spinner 20 times and record the number of times it lands on red.\\
(a) Find the critical region for Sam's test.\\
(b) Write down the size of Sam's test.
Tessa decides to spin the spinner until it lands on red and she records the number of spins.\\
(c) Find the critical region for Tessa's test.\\
(d) Find the size of Tessa's test.\\
(e) (i) Show that the power function for Sam's test is given by
$$( 1 - p ) ^ { 19 } ( 1 + 19 p )$$
(ii) Find the power function for Tessa's test.\\
(f) With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when $\mathrm { p } = 0.15$\\
\hfill \mbox{\textit{Edexcel FS1 Q7 [18]}}